Q-systems and extensions of completely unitaryvertex operator algebras

BIN GUI

Contents

0 Introduction 2

1 Intertwining operators and tensor categories of unitary VOAs 51.1 Braiding, fusion, and contragredient intertwining operators . . . . . . . 51.2 Unitary VOAs and unitary representations . . . . . . . . . . . . . . . . . 91.3 Adjoint and conjugate intertwining operators . . . . . . . . . . . . . . . . 10

2 Unitary VOA extensions, C�-Frobenius algebras, and their representations 142.1 Preunitary VOA extensions and commutative C-algebras . . . . . . . . . 142.2 Duals and standard evaluations in C�-tensor categories . . . . . . . . . . 162.3 Unitarity of C-algebras and VOA extensions . . . . . . . . . . . . . . . . . 192.4 Unitary C-algebras and C�-Frobenius algebras . . . . . . . . . . . . . . . 212.5 Strong unitarity of unitary VOA extensions . . . . . . . . . . . . . . . . . 25

3 C�-tensor categories associated to Q-systems and unitary VOA extensions 293.1 Unitary tensor products of unitary bimodules of Q-systems . . . . . . . . 293.2 C�-tensor categories associated to Q-systems . . . . . . . . . . . . . . . . 323.3 Dualizable unitary bimodules . . . . . . . . . . . . . . . . . . . . . . . . . 373.4 Braiding and ribbon structures . . . . . . . . . . . . . . . . . . . . . . . . 403.5 Complete unitarity of unitary VOA extensions . . . . . . . . . . . . . . . 433.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Index 50

References 51

1

Abstract

Complete unitarity is a natural condition on a CFT-type regular VOA whichensures that its modular tensor category is unitary. In this paper we show thatany CFT-type unitary (conformal) extension U of a completely unitary VOA Vis completely unitary. Our method is to relate U with a Q-system AU in the C�-tensor category RepupV q of unitary V -modules. We also update the main result of[KO02] to the unitary cases by showing that the tensor category RepupUq of unitaryU -modules is equivalent to the tensor category RepupAU q of unitary AU -modulesas unitary modular tensor categories.

As an application, we obtain infinitely many new (regular and) completelyunitary VOAs including all CFT-type c   1 unitary VOAs. We also show that thelatter are in one to one correspondence with the (irreducible) conformal nets of thesame central charge c, the classification of which is given by [KL04].

0 Introduction

This is the first part in a series of papers to study the relations between unitary ver-tex operator algebra (VOA) extensions and conformal net extensions. We will alwaysfocus on rational conformal field theories, so our VOAs are assumed to be CFT-type,self-dual, and regular, so that the categories of VOA modules are modular tensor cat-egories (MTCs).

Although both unitary VOAs and conformal nets are mathematical formulations ofunitary chiral CFTs, they are defined and studied in rather different ways, with the for-mer being more algebraic and geometric, and the latter mainly functional analytic. Asystematic study to relate these two approaches was initiated by Carpi-Kawahigashi-Longo-Weiner [CKLW18] and followed by [Ten19a, Ten18b, Gui20, Ten19b, CW,CWX], etc. In these works the methods of relating the two approaches are transcen-dental and have a lot of analytic subtleties. Due to these subtleties, certain models(such as unitary Virasoro VOAs, and unitary affine VOAs especially of type A) areeasier to analyze than the others. On the other hand, when studying the extensionsand conformal inclusions of chiral CFTs, the main tools in the two approaches are quitesimilar: both are (commutative) associative algebras in a tensor category C (called C-algebras); see [KO02, HKL15, CKM17] for VOA extensions, and [LR95, KL04, BKLR15]for conformal net extensions; see also [FRS02, FS03] for the general notion of algebraobjects inside a tensor category. In this and the following papers, we will see that C-algebras are also powerful tools for relating unitary VOA extensions and conformalnet extensions in the above mentioned systematic and transcendental settings.

There is, however, one important difference between the C-algebras used in the twoapproaches: for conformal net extensions the C-algebras are unitary. Unitarity is anessential property for conformal nets and operator algebras but not quite necessary forVOAs. However, it is impossible to relate VOAs and conformal nets without addingunitary structures on VOAs (and their representation categories). This point is alreadyclear in [Gui20], where we have seen that to relate the tensor categories of VOAs andconformal nets, one has to first make the VOA tensor categories unitary.

In this paper our main goal is to relate the C�-tensor categories of unitary VOAextensions with those of unitary C-algebras (also called C�-Frobenius algebras or (un-

2

der slightly stronger condition) Q-systems [Lon94]). As applications, we prove manyimportant unitary properties of VOA extensions, the most important of which are thecomplete unitarity of VOAs as defined below.

Complete unitarity of unitary VOA extensions

A CFT-type regular VOA V is called completely unitary if the following conditionsare satisfied.

• V is unitary [DL14], which means roughly that V is equipped with an inner prod-uct and an antiunitary antiautomorphism Θ which relates the vertex operatorsof V to their adjoints.

• Any V -module admits a unitary structure. Since V -modules are semisimple, itsuffices to assume the unitarizability of irreducible V -modules.

• For any irreducible unitary V -modules Wi,Wj,Wk, the non-degenerate invariantsesquilinear form Λ introduced in [Gui19b] and defined on the dual vector spaceof type

�ki j

�intertwining operators of V is positive.

The importance of complete unitarity lies in the following theorem.

Theorem 0.1 ([Gui19b] theorem 7.9). If V is a CFT-type, regular, and completely unitaryVOA, then the unitary V -modules form a unitary modular tensor category.

However, compared to unitarity, complete unitarity is much harder to prove sincenot only vertex operators but also intertwining operators need to be taken care of.In this paper, our main result as follows provides a powerful tool for proving thecomplete unitarity.

Theorem 0.2. Suppose that V is a CFT-type, regular, and completely unitary VOA, and U isa CFT-type unitary VOA extension of V . Then U is also completely unitary.

Roughly speaking, if we know that a unitary VOAU is an extension of a completelyunitary VOA V , then U is also completely unitary. As applications, since the completeunitarity has been established for unitary affine VOAs and c   1 unitary VirasoroVOAs (minimal models) as well as their tensor products (proposition 3.31), we knowthat all their unitary extensions are completely unitary. In particular, these extensionshave unitary modular tensor categories. We also show that these tensor categories areequivalent to the unitary modular tensor categories associated to the correspondingQ-system. To be more precise, we prove

Theorem 0.3. Let V be a CFT-type, regular, and completely unitary VOA, and let U be aCFT-type unitary extension of V whose Q-system is AU . If RepupUq is the category of unitaryU -modules, and RepupAUq is the category of unitary AU -modules, then RepupUq is natuallyequivalent to RepupAUq as unitary modular tensor categories.

A non-unitary version of the above theorem has already been proved in [CKM17]:RepupUq is known to be equivalent to RepupAUq as modular tensor categories. Theabove theorem says that the unitary structures of the two modular tensor categories

3

are also equivalent in a natural way. We remark that the unitary tensor structurescompatible with the �-structure of RepupUq are unique by [Reu19, CCP]. Thus [Reu19,CCP] provide a different method of proving the above theorem.

We would like to point out that our results on the relation between unitary VOA ex-tensions and Q-systems also provide a new method of proving the unitarity of VOAs.For instance, the irreducible c   1 conformal nets are classified as (finite index) ex-tensions of Virasoro nets by Kawahigashi-Longo in [KL04] (table 3), and their VOAcounterparts are given by Dong-Lin in [DL15]. However, Dong-Lin were not able toprove the unitarity of two exceptional cases: the types pA10, E6q and pA28, E8q (see re-mark 4.16 of [DL15]). But since these two types are realized as commutative Q-systemsin [KL04], we can now show that the corresponding VOAs are actually unitary. Thisproves that the c   1 CFT-type unitary VOAs are in one to one correspondence withthe irreducibile conformal nets with the same central charge c.

We have left several important questions unanswered in this paper. We see thatQ-systems can relate unitary VOA extensions and conformal net extensions. But[CKLW18] also provides a uniform way of relating unitary VOAs and conformal netsusing smeared vertex operators. Are these two relations compatible? Moreover, dothe VOA extensions and the corresponding conformal net extensions have the sametensor categories? Answers to these questions are out of scope of this paper, so weleave them to future works.

Outline of the paper

In chapter 1 we review the construction and basis properties of VOA tensor cat-egories due to Huang-Lepowsky. We also review various methods of constructingnew intertwining operators from old ones, and translate them into tensor categori-cal language. The translation of adjoint and conjugate intertwining operators is themost important result of this chapter. Unitary VOAs, unitary representations, and theunitary structure on VOA tensor categories are also reviewed.

In chapter 2 we relate unitary VOA extensions and Q-systems as well as their (uni-tary) representations. The first two sections serve as background materials. In section2.1 we review the relation between VOA extensions and commutative C-algebras as in[HKL15]. Their results are adapted to our unitary setting. In section 2.2 we review var-ious notions concerning dualizable objects in C�-tensor categories. Most importantly,we review the construction of standard evaluations and coevaluations in C�-tensorcategories necessary for defining quantum traces and quantum dimensions. Standardreference for this topic is [LR97]. We also explain why the naturally defined eval-uations and coevaluations in the tensor categories associated to completely unitaryVOAs are standard. In section 2.3 we define a notion of unitary C-algebras which is adirect translation of unitary VOA extensions in categorical language. This notion is re-lated to C�-Frobenius algebras and Q-systems in section 2.4. The equivalence of c   1unitary VOAs and conformal nets is also proved in that section. A VOA U is calledstrongly unitary if it satisfies the first two of the three conditions defining completeunitarity. Therefore, strong unitarity means the unitarity of U and the unitarizabilityof all U -modules. In section 2.5, we give two proofs that any unitary extension U of acompletely unitary VOA V is strongly unitary. The first proof uses induced represen-

4

tations, and the second one uses a result of standard representations of Q-systems in[BKLR15].

In chapter 3 we use theC�-tensor categories of the bimodules of Q-systems to provethe complete unitarity of unitary VOA extensions. We review the construction andbasic properties of these C�-tensor categories in the first four sections. Although theseresults are known to experts (cf. [NY16] chapter 6 or [NY18b] section 4.1), we providedetailed and self-contained proofs of all the relevant facts, which we hope are helpfulto the readers who are not familiar with tensor categories. We present the theory insuch a way that it can be directly compared with the (Hermitian) tensor categories ofunitary VOA modules. So in some sense our approach is closer in spirit to [KO02,CKM17]. In section 3.5 we prove the main results of this paper: theorems 0.2 and 0.3(which are theorem 3.30 of that section). Finally, applications are given in section 3.6.

Acknowledgment

I’m grateful to Robert McRae for a helpful discussion of the rationality of VOAextensions; Marcel Bischoff for informing me of certain literature on conformal inclu-sions; and Sergey Neshveyev and Makoto Yamashita for many discussions and helpfulcomments.

1 Intertwining operators and tensor categories of unitaryVOAs

1.1 Braiding, fusion, and contragredient intertwining operators

Let V be a self-dual VOA with vacuum vector Ω and conformal vector ν. Forany v P V , its vertex operator is written as Y pv, zq � °nPZ Y pvqnz�n�1. ThentLn � Y pνqn�1 : n P Zu are the Virasoro operators. Throughout this paper, we as-sume that the grading of V satisfies V � CΩ ` �ÀnPZ¡0 V pnq� where V pnq is theeigenspace of L0 with eigenvalue n, i.e., V is of CFT-type. We assume also that Vis regular, which is equivalent to that V is rational and C2-cofinite. (See [DLM97] forthe definition of these terminologies as well as the equivalence theorem.) Such con-dition guarantees that the intertwining operators of V satisfy the braiding and fusionrelations [Hua95, Hua05a] and the modular invariance [Zhu96, Hua05b], and that thecategory ReppV q of (automatically semisimple) V -modules is indeed a modular tensorcategory [Hua08].

We refer the reader to [BK01, EGNO, Tur] for the general theory of tensor cate-gories, and [Hua08] for the construction of the tensor category ReppV q of V -modules.A brief review of this construction can also be found in [Gui19a] section 2.4 or [Gui20]section 4.1. Here we outline some of the key properties of ReppV q which will be usedin the future.

Representations of V are written as Wi,Wj,Wk, etc. If a V -module Wi is given, itscontragredient module (cf. [FHL93] section 5.2) is written as Wi . Wi, the contragredi-

ent module of Wi, can naturally be identified with Wi. So we write i � i. Note that thesymbol i is now reserved for representations. So we write the imaginary unit

?�1 as

5

i. Let W0 be the vacuum V -module, which is also the identity object in ReppV q. Theproduct of V -modules is constructed in such a way that for any Wi,Wj,Wk there is acanonical isomorphism of linear spaces

Y : HompWi bWj,Wkq �ÝÑ V�k

i j

, α ÞÑ Yα (1.1)

where V�ki j

� � V� WkWiWj

�is the (finite-dimensional) vector space of intertwining op-

erators of V . For any wpiq P Wi, we write Yαpwpiq, zq �°sPC Yαpwpiqqsz�s�1 where z

is a complex variable defined in C� :� Czt0u and Yαpwpiqqs : Wj Ñ Wk is the s-thmode of the intertwining operator. We say that Wi,Wj,Wk are respectively the chargespace, the source space, and the target space of the intertwining operator Yα. Tensorproducts of morphisms are defined such that the following condition is satisfied: IfF P HompWi1 ,Wiq, G P HompWj1 ,Wjq, K P HompWk,Wk1q, then for any wpi1q P Wi1 ,

YKαpFbGqpwpi1q, zq � KYαpFwpi1q, zqG. (1.2)

One way to realize the above properties is as follows: Notice first of all that Vhas finitely many equivalence classes of irreducible unitary V -modules. Fix, for eachequivalence class, a representing element, and let them form a finite set E . We assumethat the vacuum unitary module V � W0 is in E . If Wt P E , we will use the notationt P E to simplify formulas. We then define Wi b Wj to be

ÀtPE V

�ti j

�� b Wt [HL95](here V

�ti j

�� is the dual vector space of V� ti j

�). Then for each t P E there is a natural

identification between HompWi b Wj,Wtq and V�ti j

�, which can be extended to the

general case HompWi bWj,Wkq � V�ki j

�via the canonical isomorphisms

HompWi bWj,Wkq �àtPE

HompWi bWj,Wtq b HompWt,Wkq, (1.3)

V�k

i j

�à

tPEV�t

i j

b HompWt,Wkq. (1.4)

The tensor structure of ReppV q is defined in such a way that it is related to thefusion relations of the intertwining operators of V as follows. Choose non-zero z, ζwith the same arguments (notation: arg z � arg ζ) satisfying 0   |z � ζ|   |ζ|   |z|.In particular, we assume that z, ζ are on a common ray stemming from the origin. Wealso choose argpz � ζq � arg ζ � arg z. Suppose that we have Wi,Wj,Wk,Wl,Wp,Wqin ReppV q, and intertwining operators Yα P V

�li p

�,Yβ P V

�pj k

�,Yγ P V

�qi j

�,Yδ P V

�lq k

�,

such that for any wpiq P Wi, wpjq P Wj , the following fusion relation holds when actingon Wk:

Yαpwpiq, zqYβpwpjq, ζq � Yδ�Yγpwpiq, z � ζqwpjq, ζ

�. (1.5)

Then, under the identification of Wi b pWj b Wkq and pWi b Wjq b Wk (which wedenote by Wi b Wj b Wk) via the associativity isomorphism, we have the identity

6

αp1i b βq � δpγ b 1kq, which can be expressed graphically asi j k

l

p β

α

�

i j k

l

qγ

δ . (1.6)

Here we take the convention that morphisms go from top to bottom.

Convention 1.1. When we consider fusion relations in the form (1.5), we always as-sume 0   |z � ζ|   |ζ|   |z| and argpz � ζq � arg ζ � arg z.

The braided and the contragredient intertwining operators are two major ways ofconstructing new intertwining operators from old ones [FHL93]. As we shall see, theycan all be translated into operations on morphisms. We first discuss braiding. GivenYα P V

�ki j

�, we can define braided intertwining operators B�Yα, B�Yα of type V

�kj i

�in the following way: Choose any wpiq P Wi, wpjq P Wj . Then

pB�Yαqpwpjq, zqwpiq � ezL�1Yαpwpiq, e�iπzqwpjq. (1.7)

Then the braid isomorphism ß � ßi,j : Wi bWj Ñ Wj bWi is constructed in such away that B�Yα � Yα�ß�1 . Write B�Yα � YB�α and B�Yα � YB�α. Then B�α � α � ß�1.Using and to denote ß and ß�1 respectively, this formula can be pictured as

j i

k

B�α �

j i

k

α,

j i

k

B�α �

j i

k

α. (1.8)

Let Yi � Yipv, zq be the vertex operator associated to the module Wi. Then Yi is alsoa type

�i

0 i

�intertwining operator. It’s easy to verify that B�Yi � B�Yi as type V

�ii 0

�intertwining operators, which we denote by Yκpiq and call the creation operator of Wi.Then the canonical isomorphism of the left multiplication by identity W0bWi

�Ñ Wi isdefined to be the one corresponding to Yi. Similarly the right multiplication by identityWi bW0

�Ñ Wi is chosen to be κpiq.One can also construct contragredient intertwining operators C�Yα �

YC�α, C�Yα � YC�α of Yα, which are of type�j

i k

�, such that for any wpiq P Wi, wpjq P

Wj, wpkq P Wk,

xYC�αpwpiq, zqwpkq, wpjqy � xwpkq,YαpezL1peiπz�2qL0wpiq, z�1qwpjqy. (1.9)

Here, and also throughout this paper, we follow the convention arg zr � r arg z (r P R)unless otherwise stated. To express contragredient intertwining operators graphically,we first introduce, for any V -module Wi (together with its contragredient module Wi),

7

two important intertwining operators Yevi,i P V�

0i i

�and Yevi,i P V

�0i i

�, called the an-

nihilation operators of Wi and Wi respectively. Recall that V is self dual. Fix an iso-morphism V � W0 � W0 and identify W0 and W0 through this isomorphism. We nowdefine

Yevi,i � C�Yκpiq � C�B�Yi. (1.10)

The type of Yevi,i shows that evi,i P HompWi b Wi, V q, which plays the role of theevaluation map of Wi. evi,i P HompWi bWi, V q can be defined in a similar way. Wewrite evi,i � and evi,i � � , following the convention that a verticalline with label i but upward-pointing arrow means (the identity morphism of) Wi. Wecan now give a categorical description of C�α with the help of the following fusionrelation (cf. [Gui19b] remark 5.4)

Yevj,jpwpjq, zqYC�αpwpiq, ζq � Yevk,k�YB�αpwpjq, z � ζqwpiq, ζ

�, (1.11)

which can be translated toj i k

C�α

j

�

j i k

α. (1.12)

By [Hua08] section 3, there exist coevaluation maps coevi,i � P HompV,Wi bWiqand coevi,i � � P HompV,Wi bWiq satisfying

pevi,i b 1iqp1i b coevi,iq � 1i � p1i b evi,iqpcoevi,i b 1iq, (1.13)pevi,i b 1iqp1i b coevi,iq � 1i � p1i b evi,iqpcoevi,i b 1iq. (1.14)

Thus, let Wj tensor both sides of equation (1.12) from the left, and then apply coevj,j b1i b 1k to the tops, we obtainProposition 1.2. For any V -modules Wi,Wj,Wk, and any Yα P V

�ki j

�,

i k

C�α

j

�

i k

α

j

. (1.15)

Finally, we remark that the ribbon structure on ReppV q is defined by the twist ϑ �ϑi :� e2iπL0 P EndpWiq for any V -module Wi.

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1.2 Unitary VOAs and unitary representations

In this section, we only assume that V is of CFT-type, and discuss the unitary con-ditions on V and its tensor category. We do not assume, at the beginning, that V isself-dual. In particular, we do not identify W0 with W0. As we shall see, the unitarystructure on V is closely related to certain V -module isomorphism � : W0 Ñ W0.

Suppose V is equipped with a normalized inner product x�|�y (“normalized” meansxΩ|Ωy � 1). We say that pV, x�|�yq (simply V in the future) is unitary [DL14, CKLW18], ifthere exists an anti-unitary anti-automorphism Θ of V (called PCT operator) satisfyingthat

xY pv, zqv1|v2y � xv1|Y pezL1p�z�2qL0Θv, z�1qv2y. (1.16)We abbreviate the above relation to

Y pv, zq � Y pezL1p�z�2qL0Θv, z�1q:, (1.17)with : understood as formal adjoint.

By our definition, Θ : V Ñ V is an antilinear (i.e., conjugate linear) bijective mapsatisfying

xu|vy � xΘv|Θuy, (1.18)ΘY pu, zqv � Y pΘu, zqΘv (1.19)

for all u, v P V . Such Θ is unique by [CKLW18] proposition 5.1. Note that Θv �ΘY pΩ, zqv � Y pΘΩ, zqΘv which implies ΘΩ � Ω. Also Θν � ν by [CKLW18] corollary4.11. Indeed, by that corollary, under the assumption of anti-automorphism, anti-unitarity is equivalent to Θν � ν, and also equivalent to that Θ preserves the gradingof V . Note also that Θ is uniquely determined by V and its inner product x�|�y, and thatΘ2 � 1V ([CKLW18] proposition 5.1).

Later we will discuss the unitarity of VOA extensions using tensor-categoricalmethods. For that purpose the map Θ is difficult to deal with due to its anti-linearity.So let us give an equivalent description of unitarity using linear maps. Let V � be thedual vector space of V . The inner product x�|�y on V induces naturally an antilinearinjective map A : V Ñ V �, v ÞÑ x�|vy. We set V � AV , and define an inner product onV , also denoted by x�|�y, under which A becomes anti-unitary. Equivalently, we requirexu|vy � xAv|Auy for all u, v P V . The adjoint (which is also the inverse) of A : V Ñ V isdenoted by A : V Ñ V . The reason we use the same symbol for the two conjugationmaps is to regard A as an involution. We also adopt the notation v � Av and v � v. Wewill use both v and A for conjugation very frequently, but the later is used more oftenwhen no specific vectors are mentioned.

Proposition 1.3. pV, x�|�yq is unitary if and only if there exists a (unique) unitary map � :V Ñ V , called the reflection operator of V , such that the following two relations hold for allv P V :

Y pv, zq � Y pezL1p�z�2qL0�v, z�1q:, (1.20)�Y p��v, zq�� � AY pv, zqA. (1.21)

When V is unitary, we have

A� � ��A. (1.22)

9

Note that the first and the third equations are acting on V , while the second one onV .

Proof. � and Θ are related by � � AΘ. Then �v � Θv, ��v � Θ�1v. It is easy to see that(1.20) and (1.21) are equivalent to (1.17) and (1.19) respectively. The uniqueness of �follows from that of Θ. When V is unitary, A� � Θ � Θ�1 � ��A.

In the following we always assume V to be unitary. Let Wi be a V -module,and assume that the vector space Wi is equipped with an inner product x�|�y. Wesay that pWi, x�|�yq (or just Wi) is a unitary V -module, if for any v P V we haveYipv, zq � YipezL1p�z�2qL0Θv, z�1q: when acting on Wi. Equivalently,

Yipv, zq � YipezL1p�z�2qL0�v, z�1q:. (1.23)A V -module is called unitarizable if it can be equipped with an inner product underwhich it becomes a unitary V -module. An intertwining operator Yα of V is calledunitary if it is among unitary V -modules.

Note that just as the conjugations between V and V , given a unitary V -module Wi,we have a natural conjugation A : Wi Ñ Wi whose inverse is also written as A : Wi ÑWi. (Indeed, we first have an injective antilinear map A : Wi Ñ W �i . That AWi � Wi, i.e.,that vectors in AWi are precisely those with finite conformal weights, follows from thefact that xL0 � |�y � x�|L0�y which is a consequence of (1.23) and the fact that �ν � Θν �ν.) Fix an inner product on Wi under which A becomes anti-unitary. Then Wi is alsoa unitary V -module. For any wpiq P Wi, write wpiq � Awpiq, wpiq � Awpiq � wpiq. Recallby our notation that Yi and Yi are the vertex operators of Wi and Wi respectively. Therelation between these two operators is pretty simple: YipΘv, zq � AYipv, zqA, acting onWi. (See [Gui19a] formula (1.19).) Equivalently,

Yip�v, zq � AYipv, zqA. (1.24)This equation (applied to i � 0), together with (1.21) and (1.22), shows for any v P Vthat �Y0pv, zq�� � Y0pv, zq, where Y0 � Y is the vertex operator of the vacuum moduleV , and Y0 is that of its contragredient module. We conclude:

Proposition 1.4. If V is a unitary CFT-type VOA, then V is self-dual. The vacuum moduleV � W0 is unitarily equivalent to its contragredient module V � W0 via the reflection operator�.

Convention 1.5. Unless otherwise stated, if V is unitary and CFT-type, the isomor-phism W0

�Ñ W0 is always chosen to be the reflection operator �, and the identificationof V � W0 and V � W0 is always assumed to be through �.

1.3 Adjoint and conjugate intertwining operators

We have seen in section 1.1 two ways of producing new intertwining operatorsfrom old ones: the braided and the contragredient intertwining operators. In the uni-tary case there are two extra methods: the conjugate and the adjoint intertwining op-erators. In this section, our main goal is to derive tensor-categorical descriptions ofthese two constructions of intertwining operators.

10

First we review the definition of these two constructions; see [Gui19a] section 1.3for more details and basic properties. Let V be unitary and CFT-type. Choose unitaryV -modulesWi,Wj,Wk. For any Yα P V

�ki j

�, its conjugate intertwining operator AYα �

YAα � Yα is of type V�ki j

�defined by

Yαpwpiq, zq � AYαpwpiq, zqA (1.25)

for any wpiq P Wi. Despite its simple form, one cannot directly translate this definitioninto tensor categorical language, again due to the anti-linearity of A. Therefore we needto first consider the adjoint intertwining operator Y:α � Yα: P V

�j

i k

�, defined by

α: � C�α � C�α, (1.26)which will be closely related to the �-structures of the C�-tensor categories. Then forany wpiq P Wi,

Yα:pwpiq, zq � YαpezL1pe�iπz�2qL0wpiq, z�1q:. (1.27)Note that : is an involution: α:: � α. Moreover, by unitarity, up to equivalence of thecharge spaces � : V �Ñ V , Yi is equal to its adjoint intertwining operator, and obviouslyalso equal to the conjugate intertwining operator of Yi. Another important relation is

evi,i � κpiq: (1.28)by [Gui19a] formula (1.44). Recall from section 1.1 that, up to the isomorphism W0 �W0, evi,i is defined to be C�κpiq. Due to convention 1.5, the more precise definition isevi,i � ��1 � C�κpiq. We will use (1.28) more often than this definition in our paper.

We shall now relate α: with the unitarity structure of the tensor category of unitaryV -modules. Let V be a CFT-type VOA. Assume that V is strongly unitary [Ten18a],which means that V is unitary, and any V -module is unitarizable. Then RepupV q,the category of unitary V -modules, is a C�-category, whose � structure is defined asfollows: if Wi,Wj are unitary and T P HompWi,Wjq, then T � P HompWj,Wiq is simplythe adjoint of T , defined with respect to the inner products of Wi and Wj .

Assume also that V is regular. To make RepupV q a C�-tensor category, we have tochoose, for any unitary V -modules Wi,Wj , a suitable unitary structure on Wi b Wj .Note that it is already known that Wi bWj is unitarizable by the strong unitary of V .But here the unitary structure has to be chosen such that the structural isomorphismsbecome unitary. So, for instance, if Wk is also unitary, the associativity isomorphismpWibWjqbWk �Ñ Wib pWj bWkq and the braid isomorphism ß : WibWj �Ñ Wj bWihave to be unitary. Then we can identify pWi bWjq bWk and Wi b pWj bWkq as thesame unitary V -module called Wi bWj bWk.

To fulfill these purposes, recall thatWibWj is defined to beÀ

tPE V�ti j

��bWt. Sinceevery Wt already has a unitary structure, it suffices to assume that the direct sum isorthogonal, and define a suitable inner product Λ (called invariant inner product) oneach V

�ti j

��.Let us first assume that we can always find Λ which make RepupV q a braided C�-

tensor category. Then we can define an inner product on V�ti j

�, also denoted by Λ,

11

under which the anti-linear map V�ti j

� Ñ V� ti j

��, Yα ÞÑ Λp�|Yαq becomes anti-unitary.Here we assume Λp�|�q to be linear on the first variable and anti-linear on the secondone. Let xYα : α P Ξti,jy be a basis of V

�ti j

�, and let x qYα : α P Ξti,jy be its dual basis in

V�ti j

��. In other words we assume for any α, β P Ξti,j that xYα, qYβy � δα,β . The follow-ing proposition relates Λ with the categorical inner product.

Proposition 1.6. For any Yα,Yβ P V�ti j

�we have

αβ� � ΛpYα|Yβq1t. (1.29)

Proof. Assume that Ξti,j is orthonormal under Λ. Then so is x qYα : α P Ξti,jy. For anyα P Ξti,j , let Pα be the projection mapping Wi bWj �

ÀKtPE V

�ti j

�� bWt onto qYα bWt,and let Uα be the unitary map qYα bWt Ñ Wt, qYα b wptq ÞÑ wptq. Then we clearly haveα � UαPα. So αα� � 1t. If β P Ξti,j and β � α, then clearly PβPα � 0, and henceαβ� � 0. Thus (1.28) holds for basis vectors, and hence holds in general.

We warn the reader the difference of the two notations α: and α�. If Yα P V�ti j

�then α: P HompWi b Wt,Wjq, while α�, defined using the �-structure of RepupV q, isin HompWt,Wi bWjq. Thus, Yα: is the adjoint intertwining operator while Yα� makesno sense. This is different from the notations used in [Gui19a, Gui19b], where α: isnot defined but Yα� (also written as Y:α) denotes the adjoint intertwining operator.Despite such difference, α: and α�, the VOA adjoint and the categorical adjoint, shouldbe related in a natural way. And it is time to review the construction of invariant Λintroduced in [Gui19b].

Definition 1.7. Let Wi,Wj be unitary V -modules. Then the invariant sesquilinearform Λ on V

�ti j

�� for any t P E is defined such that the fusion relation holds for anywpiq1 , w

piq2 P Wi:

Yj�Yevi,ipw

piq2 , z � ζqwpiq1 , ζ

� �¸tPE

¸α,βPΞti,j

Λp qYα| qYβq � Yβ:pwpiq2 , zqYαpwpiq1 , ζq. (1.30)Equation (1.30) can equivalently be presented as

i ji

j

�¸tPE

¸α,βPΞti,j

Λp qYα| qYβqi j

j

t α

β:

i

. (1.31)

It is not too hard to show that Λ is Hermitian (i.e. Λp qYα| qYβq � Λp qYβ| qYαq). In-deed, one can prove this by applying [Gui19b] formula (5.34) to the adjoint of (1.30).(The intertwining operator Yσ̃ in that formula could be determined from the proofsof [Gui19b] corollary 5.7 and theorem 5.5.) Moreover, from the rigidity of ReppV q onecan deduce that Λ is non-degenerate; see [HK07] theorem 3.4,1 or step 3 of the proof of

1The relation between the bilinear form in [HK07] theorem 3.4 and the sesquilinear form Λ is ex-plained in [Gui19b] section 8.3.

12

[Gui19b] theorem 6.7. However, to make Λ an inner product one has to prove that Λis positive. In [Gui19b, Gui19c] we have proved the positivity of Λ for many examplesof VOAs. As mentioned in the introduction, one of our main goal of this paper is toprove that all unitary extensions of these examples have positive Λ. We first introducethe following definition.

Definition 1.8. Let V be a regular and CFT-type VOA. We say that V is completelyunitary, if V is strongly unitary (i.e., V is unitary and any V -module is unitarizable),and if for any unitary V -modulesWi,Wj and any t P E , the invariant sesquilinear formΛ defined on V

�ti j

�� is positive. In this case we call Λ the invariant inner product ofV .

As we have said, since Λ is non-degenerate, when V is completely unitary Λ be-comes an inner product on V

�ti j

�� (and hence also on V� ti j

�), which can be extended to

an inner product on WibWj , also denoted by Λ. Then WibWj becomes a unitary (butnot just unitarizable) V -module. In other words it is an object not just in ReppV q butalso in RepupV q. Moreover, as shown in [Gui19b], Λ is the right inner product whichmakes all structural isomorphisms unitary. More precisely:

Theorem 1.9 (cf. [Gui19b] theorem 7.9). If V is regular, CFT-type, and completely unitary,then RepupV q is a unitary modular tensor category.

In the remaining part of this paper, we assume, unless otherwise stated, that V isregular, CFT-type, and completely unitary. The following relation is worth noting; see[Gui19b] section 7.3.

Proposition 1.10. If Wi is unitary then coevi,i � ev�i,i.

We are now ready to state the main results of this section.

Theorem 1.11. For any unitary V -modules Wi,Wj,Wk and any Yα P V�ki j

�, we have α: �

pevi,i b 1jqp1i b α�q. In other words,

j

i k

α: �j

i k

α� . (1.32)

Proof. It suffices to assume Wk to be irreducible. So let us prove (1.32) for all k � t P Eand any basis vector α P Ξti,j . Here we assume Ξti,j to be orthonormal under Λ. Chooserα P HompWibWj,Wtq such that rα: equals pevi,ib1jqp1ibα�q. We want to show rα � α.

By proposition 1.6 we have αβ� � δα,β1t for any α, β P Ξti,j . This impliesi j

�¸tPE

¸αPΞti,j

i j

ji

tα

α� . (1.33)

13

Tensor 1i from the left and apply evi,i b 1j to the bottom, we obtain

i i j

�¸tPE

¸αPΞti,j

i i j

t

j

α

rα: , (1.34)

which, together with equation (1.31), implies

rα: � ¸βPΞti,j

Λp qYα| qYβqβ: � ¸βPΞti,j

δα,ββ: � α:.

Thus rα � α.Corollary 1.12. For any unitary V -modules Wi,Wj,Wk and any Yα P V

�ki j

�,

i j

α

k

�

i j

k

α� . (1.35)

Proof. We have α � C�C�α � pC�αq:. By propositions 1.2, 1.10, and the unitarity of ß,pC�αq� equals

α�

ki

j

. (1.36)

By theorem 1.11, one obtains pC�αq: from pC�αq� by bending the leg i to the top. Thus(1.36) becomes the right hand side of (1.35).

2 Unitary VOA extensions, C�-Frobenius algebras, andtheir representations

2.1 Preunitary VOA extensions and commutative C-algebrasIn this chapter, we fix a regular, CFT-type, and completely unitary VOA V . W0 �

V is identified with W0 � V via the reflection operator �. Thus � � 1. Let U bea (VOA) extension of V , whose vertex operator is denoted by Yµ. By definition, Uand V share the same conformal vector ν and vacuum vector Ω, and V is a vertex

14

operator subalgebra of U . As in the previous chapter, the symbol Y is reserved for thevertex operator of V . Then Y is the restriction of the action Yµ : U ñ U to V ñ V .More generally, let Ya be the restriction of Yµ to V ñ U . Then pU, Yaq becomes arepresentation of V . We write this V -module as pWa, Yaq, or Wa for short. (So by ournotation, U equals Wa as a vector space.) Since all V -modules are unitarizable, we fixa unitary structure (i.e., an inner product) x�|�y on the V -module Wa whose restrictionto V is the one x�|�y of the unitary VOA V . Such pU, x�|�yq, or U for short, is called apreunitary (VOA) extension of V . It is clear that any extension of V is preunitarizable.Finally, we notice that Yµ is a type

�aa a

� � � WaWaWa

�unitary intertwining operator of V .

The above discussion can be summarized as follows: The preunitary VOA ex-tension U is a unitary V module pWa, Yaq; its vertex operator Yµ is in V

�aa a

�. Thus

µ � P HompWa bWa,Waq by our notation in the last chapter. Let ι : W0 Ñ Wadenote the embedding of V into U . Then clearly ι P HompW0,Waq. We write ι � .Set AU � pWa, µ, ιq. Then AU is a commutative associative algebra in RepupV q (orcommutative RepupV q-algebra for short) [Par95, KO02], which means:

• (Associativity) µpµb 1aq � µp1a b µq.• (Commutativity) µ � ß � µ.• (Unit) µpιb 1aq � 1a.

Note that the associators and the unitors of RepupV q have been suppressed to simplifydiscussions. We will also do so in the remaining part of this article.

Recall that ß is the braid isomorphism ßa,a : Wa b Wa Ñ Wa b Wa. These threeconditions can respectively be pictured as

a a a

a

�a a a

a

,

a a

a

�a a

a

, a

a

a

�a

.

Indeed, associativity and commutativity are equivalent to the Jacobi identity for Yµ.The unit property follows from the fact that Yµ restricts to Ya. See [HKL15] for moredetails. Note that we also have

µ � ßn � µ, µp1a b ιq � 1a (2.1)

for any n P Z. The first equation follows from induction and that µ � µßß�1 � µß�1.The second equation holds because µp1abιq � µßa,ap1abιq � µpιb1aqßa,0 � 1aßa,0 � 1a.

Recall that the twist is defined by e2iπL0 . Since L0 has only integral eigenvalueson Wa, the unitary V -module Wa has trivial twist: ϑa � 1a. We also notice that A isnormalized (i.e., ι�ι � 10) since the inner product on U restricts to that of V . We thusconclude: If U is a preunitary extension of V thenAU is a normalized commutative RepupV q-algebra with trivial twist. Conversely, any such commutative RepupV q-algebra arises froma preunitary CFT-type extension. Moreover, if U is of CFT type, then AU is haploid,which means that dim HompW0,Waq � 1. Indeed, if Wi is a unitary V -submodule ofWa equivalent to W0, then the lowest conformal weight of Wi is 0, which implies that

15

Ω P Wi and hence that Wi � W0. Therefore dim HompW0,Waq � dim HompW0,W0q � 1by the simpleness of V . Conversely, if AU is haploid, then U is of CFT type providedthat any irreducible V -module not equivalent toW0 has no homogeneous vectors withconformal weight 0. This converse statement will not be used in this paper (except incorollary 2.22). We thus content ourselves with the following result.

Proposition 2.1 (cf. [HKL15] theorem 3.2). If U is a preunitary CFT-type VOA extensionof V , then AU is a normalized haploid commutative RepupV q-algebra with trivial twist.

A detailed discussion of unitary VOA extensions will be given in the followingsections. For now we first give the definition:

Definition 2.2. Let U be a CFT-type VOA extension of V , and assume that the vectorspace U is equipped with a normalized inner product x�|�y. We say that the extensionV U is unitary (equivalently, that U is a unitary (VOA) extension of V ), if x�|�yrestricts to the normalized inner product of V , and pU, x�|�yq is a unitary VOA.

A unitary VOA extension is clearly preunitary. Another useful fact is the following:

Proposition 2.3. If U is a CFT-type unitary VOA extension of V , then the PCT operator ΘUof U restricts to the one ΘV of V . In particular, V is a ΘU -invariant subspace of U .

Thus we can let Θ denote unambiguously both the PCT operators of U and of V .

Proof. By relation (1.17) and the fact that Θ2U � 1U ,Θ2V � 1V , for any v P V U wehave

Y pΘUv, zq � Y pezL1p�z�2qL0v, z�1q: � Y pΘV v, zq

when evaluating between vectors in V . It should be clear to the reader how the condi-tion that the normalized inner product of U restricts to that of V is used in the aboveequations. Thus ΘU |V � ΘV .

2.2 Duals and standard evaluations in C�-tensor categories

Dualizable objects

Let C be a C�-tensor category (cf. [Yam04]) whose identity object W0 is simple.We assume tacitly that C is closed under finite orthogonal direct sums and orthogonalsubojects, which means that for a finite collection tWs : s P Su of objects in C thereexists an object Wi and partial isometries tus P HompWi,Wsq : s P Su satisfying utu�s �δs,t1s (@s, t P S) and

°sPS u

�sus � 1i, and that for any object Wi and a projection p P

EndpWiq there exists an object Wj and a partial isometry u P HompWi,Wjq such thatuu� � 1j, u�u � p.2

Assume that an object Wi in C has a right dual Wi, which means that there existevaluation evi P HompWibWi,W0q and coevaluation coevi P HompW0,WibWiq satisfy-ing p1ibeviqpcoevib1iq � 1i and pevib1iqp1ibcoeviq � 1i. Set evi,i � evi, coevi,i � coevi,

2A morphism u P HompWi,Wjq is called a partial isometry if u�u and uu� are projections. A mor-phism e P EndpWiq is called a projection if e2 � e � e�.

16

and set also evi,i � pcoevi,iq�, coevi,i � pevi,iq�. Then equations (1.13) and (1.14) aresatisfied, which shows that Wi is also a left dual of Wi. In this case we say that Wiis dualizable. Note that evi,i determines the remaining three ev and coev. In gen-eral, we say that evi,i P HompWi bWi,W0q, evi,i P HompWi bWi,W0q are evaluations(or simply ev) of Wi and Wi if equations (1.13) and (1.14) are satisfied when settingcoevi,i � ev�i,i, coevi,i � ev�i,i. In the case that Wi is self-dual, we say that evi,i is anevaluation (ev) of Wi, if, by setting coevi,i � ev�i,i, we have pevi,ib 1iqp1ib coevi,iq � 1i.Taking adjoint, we also have p1i b evi,iqpcoevi,i b 1iq � 1i.

Assume that Wi,Wj are dualizable with duals Wi,Wj respectively. Choose eval-uations evi,i, evi,i, evj,j, evj,j . Then Wibj :� Wi b Wj is also dualizable with a dualWjbi :� Wj bWi and evaluations

evibj,jbi � evi,ip1i b evj,j b 1iq, evjbi,ibj � evj,jp1j b evi,i b 1jq. (2.2)Convention 2.4. Unless otherwise stated, if the ev for Wi,Wi and Wj,Wj are chosen,then we always define the ev for Wibj,Wjbi using equations (2.2).

Using ev and coev for Wi,Wj and their duals Wi,Wj , one can define for any F PHompWi,Wjq a pair of transposes F_, _F by

F_ � pevj,j b 1iqp1j b F b 1iqp1j b coevi,iq,_F � p1i b evj,jqp1i b F b 1jqpcoevi,i b 1jq.

Pictorially,

F_ � , _F � . (2.3)

One easily checks that _pF_q � F � p_F q_, pFGq_ � G_F_, _pFGq �p_Gqp_F q, pF_q� � _pF �q.

Standard evaluations

The evaluations defined above are not unique even up to unitaries. For anyevi,i, evi,i of Wi,Wi, and any invertible K P EndpWiq, revi,i :� evi,ipK b 1iq andrevi,i :� evi,ip1i b pK�q�1q are also evaluations. Thus one can normalize evaluationsto satisfy certain nice conditions. In C�-tensor categories, the ev that attract most in-terest are the so called standard evaluations. It is known that for dualizable objects,standard ev always exist and are unique up to unitaries, and that the two transposesdefined by standard ev are equal. We refer the reader to [LR97, Yam04, BDH14] forthese results. In the following, we review an explicit method of constructing standardevaluations following [Yam04]. Since the evaluations for VOA tensor categories de-fined in the previous chapter can be realized by this construction, these evaluationsare standard (see proposition 2.5).

Define scalars TrLpF q and TrRpF q for each F P EndpWiq such that evi,ipF b1iqcoevi,i � TrLpF q10 and evi,ip1i b F qcoevi,i � TrRpF q10. Then TrL is a positive linear

17

functional on EndpWiq. Moreover, TrL is faithful: if TrLpF �F q � 0, then evi,i � pF b 1iqis zero (since its absolute value is zero). So F � 0. Similar things can be said aboutTrR. We say that evi,i is a standard evaluation if TrLpF q � TrRpF q for all F P EndpWiq.Since TrLpF_q � TrRpF q � TrLpF_q by easy graphical calculus, it is easy to see thatevi,i is standard if and only if evi,i is so. Standard ev are unique up to unitaries: IfWi1 � Wi, T P HompWi1 ,Wiq is unitary, and revi,i1 , revi1,i are also standard, then we mayfind a unitary K P EndpWiq satisfying revi,i1 � evi,ipK b T q, revi1,i � evi,ipT b Kq. (See[Yam04] lemma 3.9-(iii).)

If Wi is simple, a standard evaluation is easy to construct by multiplying evi,i bysome nonzero constant λ (and hence multiplying evi,i by λ�1) such that TrLp1iq �TrRp1iq. In general, if Wi is dualizable and hence semisimple, we have orthogonalirreducible decomposition Wi �

ÀKsPSWs where each irreducible subobject Ws is du-

alizable (with a dual Ws). Choose partial isometries tus P HompWi,Wsq : s P Suand tvs P HompWi,Wsq : s P Su satisfying utu�s � δs,t1s, vtv�s � δs,t1s and

°s u

�sus �

1i,°s v

�s vs � 1i. Then we define

evi,i �¸sPS

evs,spus b vsq, evi,i �¸sPS

evs,spvs b usq (2.4)

(where evs,s and evs,s are standard for all s), define coev using adjoint. Then evi,i andevi,i are standard. (See [Yam04] lemma 3.9 for details.)3

Proposition 2.5. If V is a regular, CFT-type, and completely unitary VOA, then the ev andcoev defined in chapter 1 (same as those in [Gui19a, Gui19b]) for any object Wi in RepupV qand its contragredient module Wi are standard.

Proof. First of all, assume Wi is irreducible. By [Gui19b] proposition 7.7 and the para-graph before that, we have TrLp1iq � di � di � TrRp1iq. So evi,i is standard. Now, as-sumeWi is semisimple with finite orthogonal irreducible decompositionWi �

ÀKs Ws.

For each irreducible summand Ws, define us (resp. vs) to be the projection of Wi ontoWs (resp. Wi onto Ws). (Note that Wi and Ws are respectively the contragredient mod-ules of Wi,Ws.) Using (1.10), it is easy to see that Yevi,ipw, zq �

°s Yevs,spusw, zqvs for

any w P Wi. So the first equation of (2.4) is satisfied. Similarly, the second one of (2.4)holds.

Convention 2.6. Unless otherwise stated, for any object Wi in RepupV q, Wi is alwaysunderstood as the contragredient module of Wi, and the standard ev and coev forWi,Wi are always defined as in chapter 1.

3In [Yam04] the categories are assumed to be rigid. Thus any orthogonal subobject of Wi, i.e., anyobject Ws which is a associated with a partial isometry u : Wi Ñ Ws satisfying uu� � 1s, is dualizable.This fact is also true without assuming C to be rigid. (Cf. [ABD04] lemma 4.20.) Here is one way tosee this. Notice that we may assume TrL is tracial by multiplying evi,i by K b 1i where K is a positiveinvertible element of EndpWiq. (Cf. the proof of [ABD04] Thm. 4.12.) Thus, for any F,G P EndpWiq,we have TrLpGF q � TrLpF__ � Gq in general and TrLpFGq � TrLpGF q by tracialness, which showsF__ � F and hence F_ � _F . Thus, pF_q� � pF�q_. So F is a projection iff F_ is so. Let p � u�u.Then p_ P EndpWiq is a projection. Thus, there exist an object Ws and v P HompWi,Wsq satisfyingvv� � 1s, v

�v � p_. Then Ws is dual to Ws since one can choose evaluations evs,s :� evi,ipu� b

v�q, evs,s :� evi,ipv� b u�q.

18

It is worth noting that standardness is preserved by tensor products: If standard evare chosen for Wi,Wj and their dual Wi,Wj , then the ev of Wibj,Wjbi defined by (2.2)are also standard.

Standard ev are also characterized by minimizing quantum dimensions. Defineconstants di, di satisfying evi,icoevi,i � di10, evi,icoevi,i � di10. Then standard ev areprecisely those minimizing didi and satisfying di � di (cf. [LR97]). We will alwaysassume di, di to be those defined by standard evaluations, and call them the quantumdimensions of Wi,Wi.

2.3 Unitarity of C-algebras and VOA extensionsLet Wa be an object in C, choose µ P HompWa b Wa,Waq, ι P HompW0,Waq, and

asume that A � pWa, µ, ιq is an associative algebra in C (also called C-algebra4), whichmeans:

• (Associativity) µpµb 1aq � µp1a b µq.• (Unit) µpιb 1aq � 1a � µp1a b ιq.

Since W0 is simple, we can choose DA ¡ 0 (called the quantum dimension of A)satisfying ι�µµ�ι � DA10. We say that A is

• haploid if dim HompW0,Waq � 1;• normalized if ι�ι � 10;• special if µµ� P C1a; in this case we set scalar dA ¡ 0 such that µµ� � dA1a;• standard ifA is special,Wa is dualizable (with quantum dimension da), andDA �da.

Note that any C-algebra A is clearly equivalent to a normalized one.Assume that Wa has a dual Wa, together with (not necessarily standard) eva,a, eva,a.

Define ev for Wa bWa and Wa bWa using (2.2). Assume also that Wa is self-dual, i.e.,

Wa � Wa. Choose a unitary morphism ε � P HompWa,Waq, and write its adjoint

as ε� � . Write also µ� � , ι� � .

Definition 2.7. A unitary ε P HompWa,Waq is called a reflection operator of A (withrespect to the chosen ev of Wa,Wa), if the following two equations are satisfied:

µ � peva,a b 1aqpεb µ�q, εµpε� b ε�q � pµ�q_. (2.5)Pictorially,

a a

a

�

a a

a

, (2.6)

4In [KO02, HKL15, CKM17], commutativity is required in the definition of C-algebras when C isbraided. This is not assumed in our paper.

19

a a

a

�

a a

a

. (2.7)

Proposition 2.8. The reflection operator ε is uniquely determined by the dual Wa and the evof Wa and Wa.

Proof. Apply ι� to the bottom of (2.6), and then apply the unit property, we haveeva,apεb 1aq � ι�µ, which, by rigidity, implies

ε � pι�µb 1aqp1a b coeva,aq. (2.8)

Definition 2.9. Let C be a C�-tensor category with simple W0. A C-algebra A �pWa, µ, ιq in C is called unitary, if Wa is dualizable, and for a choice of Wa dual toWa and ev of Wa,Wa, there exists a reflection operator ε P HompWa,Waq. If, moreover,the ev of Wa,Wa are standard, we say that A is s-unitary5.

The definition of s-unitary C-algebras is independent of the choice of duals andstandard ev, as shown below:

Proposition 2.10. If A is s-unitary, then for any Wa1 dual to Wa, and any standard rev ofWa,Wa1 , there exists a reflection operator rε : Wa Ñ Wa1Proof. Since A is s-unitary, we can choose Wa dual to Wa and standard ev of Wa,Wasuch that there exists a reflection operator ε : Wa Ñ Wa. Let us define rε � pι�µ b1aqp1abcoeva,a1q and show that rε is a reflection operator. We first show that rε is unitary.Choose a unitary T P HompWa1 ,Waq. Then by the up to unitary uniqueness of standardev, there exists a unitary K P HompWa,Waq such that reva,a1 � eva,apK b T q, reva1,a �eva,apT b Kq. By (2.3), eva,apK b 1aq � eva,ap1a b K_q. Therefore reva,a1 � eva,ap1a bpK_qT q. Thus rε � pι�µ b 1aqp1a b T �pK_q�qp1a b coeva,aq � p10 b T �pK_q�qpι�µ b1aqp1a b coeva,aq, which, together with (2.8), implies rε � T �pK_q�ε. Thus rε is unitarysince ε,K_, T are unitary.

Now, from the definition of rε, we see that reva1,aprε b 1aq � ι�µ. Therefore reva1,aprε b1aq � eva,apε b 1aq. Using this fact, one can now easily check that (2.6) and (2.7) holdfor rε and the standard rev of Wa,Wa1 .

We now relate s-unitary C-algebras with unitary VOA extensions. First we need alemma.

Lemma 2.11. Let U be a preunitary CFT-type extension of V , AU � pWa, µ, ιq the associatedRepupV q-algebra, and Wa the contragredient module of Wa. Then U is a unitary VOA if andonly if there exists a unitary ε P HompWa,Waq satisfying for all wpaq P Wa that

Yµpwpaq, zq � Yµ:pεwpaq, zq, (2.9)εYµpε�wpaq, zqε� � Yµpwpaq, zq. (2.10)

5The letter “s” stands for several closely related notions: standard evaluations, spherical tensor cat-egories, symmetric Frobenius algebras [FRS02].

20

Proof. Suppose that such ε exists, then by proposition 1.3 and the definition of adjointand conjugate intertwining operators, U is unitary. Conversely, if U is unitary, then byproposition 1.3 there exists a unitary map ε : Wa Ñ Wa such that (2.9) and (2.10) aretrue. Moreover, by proposition 1.4, ε is a homomorphism of U -modules. So it is also ahomomorphism of V -modules. This proves ε P HompWa,Waq.

The following is the main result of this section.

Theorem 2.12. Let V be a regular, CFT-type, and completely unitary VOA. Let U be a CFT-type preunitary extension of V , and let AU � pWa, µ, ιq be the haploid commutative RepupV q-algebra associated to U . Then U is unitary if and only if AU is s-unitary.

Proof. Following convention 2.6, we letWa be the contragredient V -module ofWa, andchoose standard ev for Wa,Wa as in chapter 1. By lemma 2.11 and relation (1.2), theunitarity of U is equivalent to the existence of a unitary ε P HompWa,Waq satisfying

µ � µ:pεb 1aq, εµpε� b ε�q � µ. (2.11)Now, by theorem 1.11, µ: � peva,ab1aqp1abµ�q. Thus µ:pεb1aq � peva,ab1aqpεbµ�q.Therefore the first equation of (2.11) is equivalent to the first one of (2.5). By corollary1.12, µ � pßa,aµ�q_ � ppµß�1a,aq�q_, which equals pµ�q_ by the commutativity of AU .Therefore the second equation of (2.11) is also equivalent to that of (2.5). We concludethat ε satisfies (2.11) if and only if ε is a reflection operator. Thus the unitarity of Uis equivalent to the existence of a reflection operator under the standard ev, which isprecisely the s-unitarity of AU .

2.4 Unitary C-algebras and C�-Frobenius algebrasLet A � pWa, µ, ιq be a C-algebra. A is called a C�-Frobenius algebra in C if p1a b

µqpµ� b 1aq � µ�µ. By taking adjoint we have the equivalent condition µ�µ � pµ b1aqp1a b µ�q. Assume in this section that all line segments in the pictures are labeledby a. Then these two equations read

� � . (2.12)

A special (i.e. µµ� P C1a) C�-Frobenius algebra is called a Q-system.6 We remarkthat the C-algebra A is a Q-system if and only if it is special. In other words, theFrobenius relations (2.12) are consequences of the unit property, the associativity, andthe specialness of A; see [LR97] or [BKLR15] lemma 3.7.

The main goal of this section is to relate (s-)unitarity to Frobenius property. Moreprecisely, we shall show

Unitary C-algebras � C�-Frobenius algebras6We warn the reader that in the literature there is no agreement on whether standardness is required

in the definition of Q-systems. For example, the Q-systems in [BKLR15] are in fact standard Q-systemsin our paper.

21

YSpecial unitary C-algebras � Q-systems

YSpecial s-unitary C-algebras � standard Q-systems. (2.13)

Moreover, under the assumption of haploid condition, all these notions are equivalent.In the process of the proof we shall also see that (2.7) is a consequence of (2.6). Thusthe definition of reflection operator can be simplified to assume only (2.6).

To begin with, let us fix a dual Wa of Wa together with evaluations eva,a, eva,a ofWa,Wa. Choose a unitary ε P HompWa,Waq.Proposition 2.13. If ε satisfies (2.6), then eva,apεb 1aq � eva,ap1a b εq; equivalently,

� . (2.14)

Proof. Take the adjoint of (2.6) and apply pεb 1aq to the bottom, we have

� .

Bending the left legs to the top proves that µ equals

. (2.15)

We thus see that the right hand side of (2.6) equals (2.15). Finally, we apply ι� to theirbottoms and use the unit property. This proves equation (2.14).

Corollary 2.14. If ε and the evaluations eva,a, eva,a of Wa,Wa satisfy (2.6), then there ex-ists an evaluation reva,a for the self-dual object Wa such that rε :� 1a P EndpWaq and reva,aalso satisfy (2.6). Moreover, if eva,a, eva,a are standard, then one can also choose reva,a to bestandard.

Proof. We remind the reader that the definition of the evaluations of a self-dual objectis given at the beginning of section 2.2. Assuming that ε : Wa Ñ Wa satisfies (2.6), wesimply define reva,a P HompWa bWa,W0q to be the left and also the right hand side of(2.14), i.e.,

eva,a :� eva,apεb 1aq � eva,ap1a b εq. (2.16)

Then one easily checks that p reva,a b 1aqp1a bcoeva,aq � 1a where coeva,a :� p reva,aq�,and that 1a and reva,a also satisfy (2.6). If the ev for Wa,Wa are standard, then reva,a isalso standard by the unitarity of ε.

22

We shall write reva,a as eva,a instead. Then the above corollary says that when (2.6)holds, we may well assume that a � a, eva,a � eva,a (which is written as eva,a), andε � 1a. Pictorially, we may remove the arrows and the on the strings to simplifycalculations. Moreover, by (2.6) and the unit property one has eva,a � ι�µ:

� . (2.17)We now prove the main results of this section. Recall that C is a C�-tensor categorywith simple W0 and A � pWa, µ, ιq is a C-algebra.Theorem 2.15. A is a unitary C-algebra if and only if A is a C�-Frobenius algebra in C.

Theorem 2.16. If there existsWa dual toWa, evaluations eva,a, eva,a ofWa,Wa, and a unitaryε P HompWa,Waq satisfying (2.6), then ε also satisfies (2.7). Consequently, ε is a reflectionoperator, and A is unitary.

We prove the two theorems simultaneously.

Proof. Step 1. Suppose there exists a dual object Wa, evaluations of Wa,Wa, and aunitary morphism ε : Wa Ñ Wa satisfying (2.6). We assume that a � a, eva,a :� eva,a �eva,a is an evaluation of Wa, and ε � 1a. Then

� � � ,

where we have used successively equation (2.6), associativity, and again equation (2.6)in the above equations. This proves the first and hence also the second equation of(2.12). We conclude that A is a C�-Frobenius algebra.

Step 2. Assume that A is a C�-Frobenius algebra in C. We shall show that Wa isself-dual, and construct a reflection operator. Define eva,a P HompWa b Wa,W0q tobe eva,a � ι�µ (see (2.17)). One then easily verifies peva,a b 1aqp1a b peva,aq�q � 1a byapplying respectively 1abι and ι�b1a to the top and the bottom of the second equationof (2.12), and then applying the unit property. This shows that Wa is self-dual andeva,a � ι�µ is an evaluation of Wa. Therefore we can also omit arrows. Apply ι� b 1aand 1a b ι� to the bottoms of the second and the first equation of (2.12) respectively,and then use the unit property and equation (2.17), we obtain

� � , (2.18)

which proves that 1a and eva,a satisfy equation (2.6). Since the first and the third itemsof (2.18) are equal, we take the adjoint of them and bend their left or right legs to thetop to obtain

� � . (2.19)

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In other words, µ is invariant under clockwise and anticlockwise “1-click rotations”.Thus µ equals the clockwise 1-click rotation of the left hand side of (2.18), which proves(2.7) for 1a and eva,a. By (2.16), ε and the original evaluations eva,a, eva,a also satisfyequation (2.7). Therefore ε is a reflection operator of A with respect to Wa and thegiven evaluations of Wa,Wa. This finishes the proof of the two theorems.

Corollary 2.17. A is a special unitary C-algebra if and only if A is a Q-system.

Proof. Q-systems are by definition special C�-Frobenius algebras.

We now relate s-unitarity and standardness. We have seen that if A is unitarythen eva,a � ι�µ is an evaluation of Wa. Therefore, setting coeva,a � ev�a,a, we haveeva,acoeva,a � DA10 by the definition of DA. By the minimizing property of standardevaluations, we haveDA ¥ da, with equality holds if and only if eva,a is standard. Notethat 1a is a reflection operator with respect to eva,a. Therefore we have the following:

Proposition 2.18. Let A be unitary. Then eva,a :� ι�µ is an evaluation of the self-dual objectWa, and DA ¥ da. Moreover, we have DA � da if and only if A is s-unitary.Corollary 2.19. A is a special s-unitary C-algebra if and only if A is a standard Q-system.

Proof. By theorem 2.15 and the definition of Q-systems, A is a standard Q-system ifand only if A is a special unitary C-algebra satisfying DA � da. Thus the corollaryfollows immediately from the above proposition.

Thus we’ve finished proving the relations (2.13) given at the beginning of this sec-tion.

Proposition 2.20. Assuming haploid condition, the six notions in (2.13) are equivalent.

Proof. Let A be a haploid C�-Frobenius algebra in C. Then by [BKLR15] lemma 3.3, Ais special. The standardness follows from [LR97] section 6 (see also [Müg03] remark5.6-3, or [NY18a] theorem 2.9). Therefore A is a standard Q-system.

We can now restate the main result of the last section (theorem 2.12) in the follow-ing way:

Theorem 2.21. Let V be a regular, CFT-type, and completely unitary VOA. Let U be a CFT-type preunitary extension of V , and let AU � pWa, µ, ιq be the haploid commutative RepupV q-algebra associated to U . Then U is a unitary extension of V if and only if AU is a C�-Frobeniusalgebra. If this is true then AU is also a standard Q-system.

Let us give an application of this theorem.

Corollary 2.22. The c   1 CFT-type unitary VOAs are in one to one correspondence with theirreducibile conformal nets with the same central charge c. Their classifications are given by[KL04] table 3.

24

Proof. As shown in [KL04] proposition 3.5, c   1 irreducible conformal nets are pre-cisely irreducible finite-index extensions of the Virasoro net Ac with central chargec. Thus they are in 1-1 correspondence with the haploid commutative Q-systems inRepsspAcq, where RepsspAcq is the unitary modular tensor category of the semisim-ple representations of Ac. By [Gui20] theorem 5.1, RepsspAcq is unitarily equivalentto RepupV q, where V is the unitary Virasoro VOA with central charge c. By [HKL15]theorem 3.2, haploid commutative RepupV q-algebras with trivial twist are in 1-1 corre-spondence with CFT-type extensions of V . Thus, by theorem 2.21 and the equivalenceof unitary modular tensor categories, CFT-type unitary extensions of V ô haploidcommutative Q-systems in RepsspAcq. (The trivial twist condition is redundant; seetheorem 3.25.) But also CFT-type unitary extensions of V ô unitary VOAs with cen-tral charge c by [DL14] theorem 5.1. This proves the desired result.

2.5 Strong unitarity of unitary VOA extensions

Starting from this section, A � pWa, µ, ιq is assumed to be a unitary C-algebra, orequivalently, aC�-Frobenius algebra in C. We say that pWi, µLq (resp. pWi, µRq)7 is a leftA-module (resp. right A-module), if Wi is an object in C, and µL P HompWa bWi,Wiq(resp. µR P HompWi bWa,Wiq) satisfies the unit property

µLpιb 1aq resp. µRp1a b ιq � 1a (2.20)and the associativity:

µLp1a b µLq � µLpµb 1iq resp. µRpµR b 1aq � µRp1i b µq. (2.21)

We write µL � , µ�L � , µR � , µ�R � . If pWi, µLq is a leftA-module and pWi, µRq is a right A-module, we say that pWi, µL, µRq is an A-bimoduleif the following bimodule associativity holds:

µRpµL b 1aq � µLp1a b µRq. (2.22)We leave it to the reader to draw the pictures of associativity and unit property. Weabbreviate pWi, µLq, pWi, µRq, or pWi, µL, µRq to Wi when no confusion arises.

Set eva,a � ι�µ as in the last section. A left (resp. right) A-module pWi, µLq (resp.pWi, µRq) is called unitary, if

µL � peva,a b 1iqp1a b µ�Lq resp. µR � p1i b eva,aqpµ�R b 1aq. (2.23)AnA-bimodule pWa, µL, µRq is called unitary if pWa, µLq is a unitary leftA-module andpWa, µRq is a unitary right A-module. Unitarity can be stated for any evaluations andreflection operators:

Proposition 2.23. Let Wa be dual to Wa, eva,a, eva,a evaluations of Wa,a,Wa,a, and ε : Wa ÑWa a reflection operator. Then a left (resp. right) A-module pWa, µLq (resp. pWa, µRq) isunitary if and only if

µL � peva,a b 1iqpεb µ�Lq resp. µR � p1i b eva,aqpµ�R b εq. (2.24)7Later we will write µL and µR as µiL and µ

iR to emphasize the dependence of µL, µR on the Wi.

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Graphically,a i

i

�

a i

i

resp.

ai

i

�

ai

i

. (2.25)

Proof. This is obvious since we have equations (2.16).

If Wi and Wj are left (resp. right) A-modules, then a morphism F P HompWi,Wjqof C is called a left (resp. right) A-module morphism, if

µLp1a b F q � FµL resp. µRpF b 1aq � FµR, (2.26)pictorially,

a i

j

j �

a i

j

i resp.

ai

j

j �

ai

j

i . (2.27)

We let HomA,�pWi,Wjq (resp. Hom�,ApWi,Wjq) be the vector space of left (resp. right)A-module morphisms. If Wi,Wj are A-bimodules, then we set HomApWi,Wjq �HomA,�pWi,Wjq X Hom�,ApWi,Wjq to be the vector space of A-bimodule mor-phisms. In the case Wi � Wj , we write these spaces of morphisms asEndA,�pWiq,End�,ApWiq,EndApWiq respectively. Wi is called a simple or irreducibleleft A-module (resp. right A-module, A-bimodule) if EndA,�pWiq (resp. End�,ApWiq,EndApWiq) is spanned by 1i. The following proposition is worth noting:Proposition 2.24 (cf. [NY16] section 6.1). The category of unitary left A-modules (resp.right A-modules, A-bimodules) is a C�-category whose �-structure inherits from that of C. Inparticular, this category is closed under finite orthogonal direct sums and subobjects.

Proof. If Wi,Wj are unitary left A-modules and F P HomA,�pWi,Wjq, one can easilycheck that F � P HomA,�pWj,Wiq using figures (2.25) and (2.27). Hence, the C�-ness ofthe category of left A-modules follows from that of C. Existence of finite orthogonaldirect sums follow from that of C. If p P EndA,�pWiq is a projection of the unitary leftA-module pWi, µiLq, we choose an object Wk in C and a partial isometry u P HompWi,Wkqsuch that uu� � 1k, u�u � p. Then pWk, uµiLu�q is easily verified to be a unitary leftA-module. Note that the fact that p intertwines the left action of A is used to verify theassociativity.

The cases of right modules and bimodules are proved in a similar way.

A left A-module (resp. right A-module, A-bimodule) Wi is called C-dualizable ifWi is a dualizable object in C. Wi is called unitarizable if there exists a unitary left A-module (resp. right A-module, A-bimodule) Wj and an invertible F P HomA,�pWi,Wjq(resp. F P Hom�,ApWi,Wjq, F P HomApWi,Wjq).Corollary 2.25. The category of unitary C-dualizable left A-modules (resp. right A-modules,A-bimodules) is a semisimple C�-category whose �-structure inherits from that of C.

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Proof. Suppose Wi is a C-dualizable left A-module. Then EndApWiq is a C�-subalgebraof the finite dimensional C�-algebra EpWiq. Thus EndApWiq is a direct sum of matrixalgebras which implies that Wi is a finite orthogonal direct sum of irreducible left A-modules. The other types of modules are treated in a similar way.

We are now going to prove the first main result of this section, that any C-dualizablemodule is unitarizable. First we need a lemma.

Lemma 2.26. Let Wi,Wk be C-dualizable left A-modules (resp. right A-modules, A-bimodules). If Wk is unitary, and there exists a surjective F P HomA,�pWk,Wiq (resp.F P Hom�,ApWk,Wiq, F P HomApWk,Wiq), then Wi is unitarizable. In particular, Wi issemisimple as a left A-module (resp. right A-module, A-bimodule).

We remark that this lemma is obvious when Wi is already known to be semisimpleas a left, right or bi A-module, which is enough for our application to representationsof VOA extensions. (Indeed, the extension U of V considered in this paper is alwaysregular, hence its modules are semisimple.) Those who are only interested in the ap-plication to VOAs can skip the following proof.

Proof. We only prove this for left modules, since the other cases can be proved simi-larly. Write the two modules as pWi, µiLq, pWk, µkLq. Note that F �F is a positive elementin the finite dimensionalC�-algebra EndpWkq. So limnÑ8pF �F q1{n converges under theC�-norm to a projection P P EndpWkqwhich is the range projection of F �. SetG � PµkLand H � PµkLp1a b P q which are morphisms in HompWa bWk,Wkq. Then using (2.27)and the fact that F � FP , we obtain FG � FH , since both equal FµkL. ThereforepF �F qnG � pF �F qnH for any integer n ¡ 0, and hence pF �F q1{nG � pF �F q1{nH bypolynomial interpolation. Thus G � PG � PH � H . We conclude

PµkL � PµkLp1a b P q. (2.28)This equation, together with (2.25), shows p1a b P qpµkLq� � p1a b P qpµkLq�P , whoseadjoint is

µkLp1a b P q � PµkLp1a b P q. (2.29)We can therefore combine (2.28) and (2.29) to get PµkL � µkLp1a b P q. In otherwords, P is a projection in EndA,�pWkq. Thus, by proposition 2.24, one can finda unitary left A-module Wj and a partial isometry K P HomA,�pWj,Wkq satisfyingK�K � 1j and KK� � P . Therefore the left A-module Wi is equivalent to Wj sinceFK P HomA,�pWj,Wiq is invertible.Theorem 2.27. C-dualizable left A-modules, right A-modules, and A-bimodules are unitariz-able.

In particular, when C is rigid, any left A-module, right A-module, or A-bimodule isunitarizable.

Proof. For any C-dualizable object Wi in C the induced left A-module pWabWi, µb1iq,abbreviated to WabWi, is clearly C-dualizable. By the unitarity of A, one easily checksthat Wa bWi is a unitary left A-module. Now assume that pWi, µLq is a left A-module.

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Then µL P HomA,�pWa bWi,Wiq. Moreover, µL is surjective since pι b 1iqµL � 1i issurjective. Therefore Wi is unitarizable by lemma 2.26. The case of right modules isproved in a similar way. In the case that pWi, µL, µRq is a C-dualizable A-bimodule,we notice that pWa bWi bWa, µ b 1i b 1a,1a b 1i b µq is a unitary A-bimodule, andµRpµLb1aq � µLp1abµRq P HomApWabWibWa,Wiq whose surjectivity follows againfrom the unit property. Thus, again, Wi is a unitarizable A-bimodule.

In the case thatA is special, there is another proof of unitarizability due to [BKLR15]which does not require dualizability. To begin with, we say that a left A-modulepWi, µLq is standard if µLµ�L P C1i. We now follow the argument of [BKLR15] lemma3.22. For any left A-module pWi, µLq, ∆ :� pµLµ�Lq1{2 is invertible. Therefore pWi, µLqis equivalent to pWi, rµLq where rµL � ∆�1µLp1a b ∆q. Using associativity and the factthat µµ� � dA1a, one can check that pWi, rµLq is standard and rµLrµ�L � dA1i. In partic-ular, if pWi, µLq is standard then ∆ is a constant and hence rµL � µL. Thus we musthave µLµ�L � dA1i. This proves that any left A-module is equivalent to a standardleft A-module, that any standard left A-module must satisfy µLµ�L � dA1a. Moreover,by [BKLR15] formula (3.4.5), any standard left A-module is unitary. Conversely, ifpWi, µLq is unitary, one can check that µLµ�L P EndA,�pWiq and hence ∆ P EndA,�pWiq.This proves that rµL � µL and hence that pWi, µLq is standard. Right A-modulescan be proved in a similar way. When pWi, µL, µRq is an A-bimodule, we defineµLR � µRpµL b 1aq � µLp1a b µRq, and say that the A-bimodule Wi is standard ifµLRµ

�LR P C1i (cf. [BKLR15] section 3.6). With the help of ∆ :� pµLRµ�LRq1{2 one

can prove similar results as of left A-modules, with the only exception being thatµLRµ

�LR � d2A1i. We summarize the discussion in the following theorems:

Theorem 2.28. If A is a Q-system in C, then all left A-modules, right A-modules, and A-bimodules are unitarizable.

Theorem 2.29. Let A be a Q-system in C, and pWi, µLq (resp. pWi, µRq, pWi, µL, µRq) a leftA-module (resp. right A-module, A-bimodule). Then the following statements are equivalent.

• Wi is unitary.

• Wi is standard.

• µLµ�L � dA1i (left A-module case), or µRµ�R � dA1i (right A-module case), orµLRµ

�LR � d2A1i where µLR � µRpµL b 1aq � µLp1a b µRq (A-bimodule case).

If C is braided with braid operator ß, and if A is commutative, a left A-modulepWi, µLq is called single-valued if µL � µLß2 (more precisely, µL � µLßi,aßa,i). IfpWi, µLq is a single-valued left A-module, then pWi, µRq is a right A-module whereµR � µLßi,a � µLß�1a,i . Moreover, by the associativity of pWi, µLq, pWi, µL, µRq is anA-bimodule. We summarize that when C is braided and A is commutative, any single-valued left A-module is an A-bimodule. Moreover, if Wi is a unitary single-valued leftA-module, the it is also a unitary A-bimodule. The category of (unitary) single-valuedleft A-module is naturally a full subcategory of the (C�-)categories of (unitary) leftA-modules, (unitary) right A-modules, and (unitary) A-bimodules.

28

With the results obtained so far, we prove that any CFT-type unitary extension Uof V is strongly unitary. Let AU � pWa, µ, ιq be the Q-system associated to U . Supposethat Wi is a U -module with vertex operator YµL . Then Wi is also a V -module, thusme may fix an inner product x�|�y on the vector space Wi under which Wi becomes aunitary V -module. We call Wi, together with x�|�y, a preunitary U -module. Moreover,YµL can be regarded as a unitary intertwining operator of V of type

�ia i

�. Thus µL P

HompWa b Wi,Wiq. One can check that pWi, µLq is a single-valued left AU -module.Indeed, unit property is obvious, associativity follows from the Jacobi identity for YµL ,and single-valued property follows from the fact that YµLp�, zq has only integer powersof z. Conversely, any preunitary U -module arises from a single-valued AU -module.Thus the category of preunitary U -modules is naturally equivalent to the category ofsingle-valued left AU -modules. See [HKL15] for more details.

To discuss the unitarizability of U -modules, the following is needed:

Theorem 2.30. Assume that V is a CFT-type, regular, and completely unitary VOA, U is aCFT-type unitary extension of V , and Wi is a preunitary U -module. Then Wi is a unitaryU -module if and only if Wi is a unitary left AU -module. If this is true then Wi is also a unitaryAU -bimodule.

Proof. LetWa be the contragredient module ofWa, eva,a, eva,a the evaluations ofWa,Wadefined in chapter 1, and ε : U � Wa Ñ U � Wa the reflection operator with respectto the chosen dual and evaluations. By equation (1.23) and the definition of adjointintertwining operators, Wi is unitary if and only if YµLpwpaq, zq � Yµ:Lpεw

paq, zq for anywpaq P Wa � U . From equation (1.2) we know that Wi is unitary if and only if µL �µ:Lpε b 1iq. With the help of theorem 1.11, this equation is equivalent to µL � peva,a b1iqpε b µ�Lq, which by proposition 2.23 means precisely the unitarity of the left AU -module Wi. If we already have that Wi is a unitary left AU -module, then, since Wi issingle-valued, it is also a unitary A-bimodule.

We now prove the strong unitarity of U .

Theorem 2.31. If V is a CFT-type, regular, and completely unitary VOA, and U is a CFT-typeunitary extension of V , then U is strongly unitary, i.e., any U -module is unitarizable.

Proof. Since U -modules are clearly preunitarizable, we choose a preunitary U -moduleWi. Then by either theorem 2.27 or theorem 2.28, Wi is unitarizable as a left AU -module. Thus, by equation (1.2) and theorem 2.30, Wi is unitarizable as a U -module.

3 C�-tensor categories associated to Q-systems and uni-tary VOA extensions

3.1 Unitary tensor products of unitary bimodules of Q-systems

In this chapter, C is a C�-tensor category with simple W0 as before, and A �pWa, µ, ιq is a Q-system (i.e., special C�-Frobenius algebra) in C. Set evaluationeva,a � ι�µ as usual. We suppress the label a in diagram calculus. Let BIMupAq be the

29

C�-category of unitary A-bimodules whose morphisms are A-bimodule morphisms.In this and the next sections, we review the construction of a C�-tensor structure onBIMupAq. See [NY16, KO02, CKM17] for reference. Note that our setting is slightlymore general than that of [NY16], since we do not assume C is rigid or A is standard.Nevertheless, many ideas in [NY16] still work in our setting. To make our article self-contained, we include detailed proofs for all the relevant results.

Choose unitary A-bimodules pWi, µiL, µiRq, pWj, µjL, µjRq. Then Wi bWj is a unitaryA-bimodule with left action µiLb 1j and right action 1ib µjR. Define Ψi,j P HomApWibWa bWj,Wi bWjq and χi,j P EndApWi bWjq to be

Ψi,j � µiR b 1j � 1i b µjL, (3.1)χi,j � p1i b µjLqppµiRq� b 1jq � pµiR b 1jqp1i b pµjLq�q. (3.2)

Definition 3.1. LetWi,Wj be unitaryA-bimodules. We say that pWij, µi,jq (abbreviatedto Wij when no confusion arises) is a tensor product of Wi,Wj over A (cf. [CKM17]), if

• Wij � pWij, µijL , µijRq is an A-bimodule, µi,j P HomApWi bWj,Wijq,8 and µi,jΨi,j �0.

• (Universal property) If pWk, µkL, µkRq is a unitary A-bimodule, α P HomApWi bWj,Wkq, and αΨi,j � 0,9 then there exists a unique rα P HomApWij,Wkq satisfyingα � rαµi,j . In this case, we say that rα is induced by α via the tensor product Wij .

The tensor product pWij, µi,jq is called unitary ifWij is a unitaryA-bimodule and χi,j �µ�i,jµi,j .

We write µi,j � , µ�i,j � . Then the equation χi,j � µ�i,jµi,j reads

i j

ji

�

i j

i j

�

i j

i j

ij , (3.3)

which is a special case of the Frobenius relations for unitary tensor products to beproved later (theorem 3.14). (Note that the first equality of (3.3) follows from the uni-tarity of Wi and Wj .)

The existence of a tensor product is clear if one assumes moreover that C is abelian:Let Wij be a cokernel of Ψi,j , and define the bimodule structure on Wij using that ofWi b Wj . Then Wij becomes a tensor product over A of Wi and Wj ; see [KO02] formore details. However, to make the tensor product unitary one has to be more carefulwhen choosing the cokernel. In the following, we proceed in a slightly different waymotivated by [BKLR15] section 3.7, and we do not require the abelianess of C. To beginwith, using µµ� � dA1a one verifies easily that χ2i,j � dAχi,j . Therefore,

8One should not confuse ib j and ij. By our notation, Wibj �Wi bWj is different from Wij .9Such α is called a categorical intertwining operator in [CKM17].

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Lemma 3.2 ([BKLR15] lemma 3.36). d�1A χi,j P EndApWi bWjq is a projection.By proposition 2.24, there exists a unitary A-bimodule Wij and a partial isometry

ui,j P HomApWi bWj,Wijq satisfying ui,ju�i,j � 1ij and u�i,jui,j � d�1A χi,j . Setting µi,j �?dAui,j , one obtains µ�i,jµi,j � χi,j (equations (3.3)) and µi,jµ�i,j � dA1ij . We now show

that pWij, µi,jq is a unitary tensor product.Proposition 3.3. Let Wij be a unitary A-bimodule, and µi,j P HomApWi bWj,Wijq. ThenpWij, µi,jq is a unitary tensor product of Wi,Wj over A if and only if µ�i,jµi,j � χi,j andµi,jµ

�i,j � dA1ij .

Proof. “If”: Since µ�i,jµi,j � χi,j and the unitarity of the A-bimodule Wij are assumed, itsuffices to show thatWij is a tensor product. Since χi,jΨi,j clearly equals 0, we compute

µi,jΨi,j � d�1A µi,jµ�i,jµi,jΨi,j � d�1A µi,jχi,jΨi,j � 0. (3.4)If Wk is a unitary A-bimodule, α P HomApWi bWj,Wkq, and αΨi,j � 0, then one canset rα � d�1A αµ�i,j and compute

rαµi,j � d�1A αµ�i,jµi,j � d�1A αχi,j � d�1A αpµiRpµiRq� b 1jq � α,where we have used αΨi,j � 0 and µiRpµiRq� � dA1i (theorem 2.29) respectively toprove the third and the fourth equalities. If there is another pα satisfying also α � pαµi,j ,then pα � d�1A pαµi,jµ�i,j � d�1A αµ�i,j � rα. Thus the universal property is checked.

“Only if”: This will be proved after the next theorem.

Theorem 3.4. Let Wi,Wj be unitary A-bimodules. Then unitary tensor products over Aof Wi,Wj exist and are unique up to unitaries. More precisely, uniqueness means that ifpWij, µi,jq and pWij, ηi,jq are unitary tensor products of Wi,Wj over A, then there exists a(unique) unitary u P HomApWij,Wijq such that ηi,j � uµi,j .Proof. Existence has already been proved. We now prove the uniqueness. Since ηi,j PHomApWi b Wj,Wijq is annihilated by Ψi,j , by the universal property for pWij, µi,jqthere exists a unique u P HomApWij,Wijq satisfying ηi,j � uµi,j . In other words u is theA-bimodule morphism induced by ηi,j . It remains to prove that u is unitary.

We first show that u is invertible. By the universal property for pWij, ηi,jq, thereexists v P HomApWij,Wijq such that µi,j � vηi,j . Thus ηi,j � uvηi,j . Therefore, uv isinduced by ηi,j via the tensor product Wij . But 1ij is clearly also induced by ηi,j viaWij . Therefore uv � 1ij . Similarly vu � 1ij . This proves that u is invertible.

We now calculate

µ�i,ju�uµi,j � η�i,jηi,j � χi,j � µ�i,jµi,j.

By the universal property, µ�i,ju�u and µ�i,j are equal since they are induced by thesame morphism via Wij . So u�uµi,j � µi,j . By the universal property again, we haveu�u � 1ij . Therefore u is unitary.Proof of the “only if” part of Proposition 3.3. By the “if” part of proposition 3.3 and theparagraph before that, there exists a unitary tensor product pWij, µi,jq satisfyingµi,jµ

�i,j � dA1ij . By uniqueness up to unitaries, this equation holds for any unitary

tensor product.

31

In the remaining part of this section, we generalize the notion of unitary tensorproduct to more than two unitary A-bimodules. For simplicity we only discuss thecase of three bimodules. The more general cases can be treated in a similar fashionand are thus left to the reader.

Choose unitary A-bimodules Wi,Wj,Wk with left actions µiL, µjL, µ

kL and right ac-

tions µiR, µjR, µ

kR respectively. Then pWibWjbWk, µiLb1jb1k,1ib1jbµkRq is a unitary

A-bimodule.

Lemma 3.5. χi,j b 1k and 1i b χj,k commute. Define χi,j,k P EndApWi bWj bWkq to betheir product. Then d�2A χi,j,k is a projection.

Proof. The commutativity of these two morphisms is verified using the commutativityof the left and right actions of Wj . Thus d�1A χi,j b 1k and d�1A 1i b χj,k are commutingprojections, whose product is therefore also a projection.

Definition 3.6. pWijk, µi,j,kq (or Wijk for short) is called a unitary tensor product ofWi,Wj,Wk over A, if

• Wijk � pWijk, µijkL , µijkR q is a unitary A-bimodule, µi,j,k P HomApWi b Wj bWk,Wijkq, and µi,j,kpΨi,j b 1kq � µi,j,kp1i b Ψj,kq � 0.

• (Universal property) If pWl, µlL, µlRq is a unitary A-bimodule, α P HomApWi bWj b Wk,Wlq, and αpΨi,j b 1kq � αp1i b Ψj,kq � 0, then there exists a uniquerα P HomApWijk,Wlq satisfying α � rαµi,j,k. In this case, we say that rα is inducedby α via the tensor product Wijk.

• (Unitarity) χi,j,k � µi,j,kµ�i,j,k.Proposition 3.7. LetWijk be a unitaryA-bimodules, and µi,j,k P HomApWibWjbWk,Wijkq.Then pWijk, µi,j,kq is a unitary tensor product of Wi,Wj,Wk over A if and only if µ�i,j,kµi,j,k �χi,j,k and µi,j,kµ�i,j,k � d2A1ijk.Theorem 3.8. Unitary tensor products of Wi,Wj,Wk exist and are unique up to unitaries.

We omit the proofs of these two results since they can be proved in a similar wayas proposition 3.3 and theorem 3.4.

3.2 C�-tensor categories associated to Q-systems

We are now ready to define the unitary tensor structure on the C�-categoryBIMupAq of unitary A-bimodules. The tensor bifunctor bA is defined as follows.For any unitary A-bimodules Wi,Wj , we choose a unitary tensor product pWij, µi,jq.Then Wi bA Wj is just the unitary A-bimodule Wij . To define tensor product of mor-phisms, we choose another pair of unitary A-bimodules Wi1 ,Wj1 , and choose anyF P HomApWi,Wi1q and G P HomApWj,Wj1q. Of course, there is also a chosen uni-tary tensor product pWi1j1 , µi1,j1q of Wi1 ,Wj1 over A. Since F bG : Wi bWj Ñ Wi1 bWj1is clearly an A-bimodule morphism, we have µi1,j1pF bGq P HomApWibWj,Wi1j1q, andone can easily show that µi1,j1pF b GqΨi,j � 0. Therefore, by universal property, thereexists a unique morphism in HomApWij,Wi1j1q, denoted by F bA G, such that

µi1,j1pF bGq � pF bA Gqµi,j. (3.5)

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This defines the tensor product of F and G in BIMupAq. We now show that bA is a �-bifunctor. Notice that F �bAG� is defined by µi,jpF �bG�q � pF �bAG�qµi1,j1 . Therefore,using pF bGq� � F � bG� we compute

pF bA Gq� � d�1A µijµ�ijpF bA Gq� � d�1A µijpF bGq�µ�i1,j1 � d�1A µijpF � bG�qµ�i1,j1�d�1A pF � bA G�qµi1,j1µ�i1,j1 � F � bA G�.

To construct associativity isomorphisms we need the following:

Proposition 3.9. Let Wi,Wj,Wk be unitary A-bimodules. Then pWpijqk, µij,kpµi,j b 1kqq andpWipjkq, µi,jkp1i b µj,kqq are unitary tensor products of Wi,Wj,Wk over A.

Note that here Wpijqk is understood as the unitary tensor product of Wij and Wkover A, and Wipjkq is understood similarly.

Proof. The two cases can be treated in a similar way. So we only prove the first one.Set µi,j,k � µij,kpµi,j b 1kq. By proposition 3.7, it suffices to prove µ�i,j,kµi,j,k � χi,j,kand µi,j,kµ�i,j,k � d2A1ijk. The second equation follows directly from that µi,jµ�i,j � dA1ijand µij,kµ�ij,k � dA1pijqk. To prove the first one, we compute (recalling that we havesuppressed the label a)

µ�i,j,kµi,j,k �

i j k

i j k

pijqkijij �

i j k

i j k

ij

ij �

i j k

i j k

ij

�

i j k

i j k

� χi,j,k.

Corollary 3.10. For any unitary A-bimodules Wi,Wj,Wk there exists a (unique) unitaryAi,j,k P HomApWpijqk,Wipjkqq satisfying

µi,jkp1i b µj,kq � Ai,j,k � µij,kpµi,j b 1kq. (3.6)

We define the unitary associativity isomorphism Wpijqk Ñ Wipjkq to be Ai,j,k.Proof. This follows immediately from the above proposition and theorem 3.8.

Proposition 3.11 (Pentagon axiom). Let Wi,Wj,Wk,Wl be unitary A-bimodules. Then

p1i bA Aj,k,lqAi,jk,lpAi,j,k bA 1lq � Ai,j,klAij,k,l (3.7)

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Proof. One can define unitary tensor products ofWi,Wj,Wk,Wl overA in a similar wayas those of three unitary A-bimodules. Moreover, using the argument of proposition3.9 one shows that pWppijqkql, µpijqk,lpµij,kpµi,j b 1kq b 1lqq is a unitary tensor product ofWi,Wj,Wk,Wl over A. We now compute

p1i bA Aj,k,lqAi,jk,lpAi,j,k bA 1lq � µpijqk,lpµij,kpµi,j b 1kq b 1lq(3.5)ùùùù p1i bA Aj,k,lqAi,jk,l � µipjkq,lpAi,j,k b 1lqpµij,kpµi,j b 1kq b 1lq� p1i bA Aj,k,lqAi,jk,l � µipjkq,lppAi,j,k � µij,kpµi,j b 1kqq b 1lq

(3.6)ùù